Suppose I have set $S = \{v_1, v_2, v_3\}$ where $v_1 = (1,0,0,0,2)$ and $v_2 = (0,1,0,1,0)$ and $v_3 = (0,0,1,1,0)$. I want to enlarge the set $S$ to be a basis for $\mathbb{R}^5$.
Systematically I know a basis for $\mathbb{R}^5$ are $(1,0,0,0,0), (0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0),(0,0,0,0,1)$, so my trial starts with $v_4= (a_1,b_1,c_1,d_1,e_1), v_5 = (a_2, b_2, c_2, d_2, e_2)$ and do linear combinations of $k_1 v_1 + k_2 v_2 + k_3 v_3 + k_4 v_4 + k_5 v_5 $ to make those 5 elementary standrad basis.
I realize this method is not good for enlarging the set. Are there any systematic ways for enlarging set to form a basis?
Best Answer
Yes, there is a systematic way. Considering J.W. Tanner's comment, let us begin by adding the standard basis vectors $e_i$ to the set $S$, and consider the set $B = \left\{ v_1, v_2, v_3, e_1,e_2,e_3,e_4,e_5 \right\}$. Now moving along from left to right, if a vector is in the span of the ones considered previously, throw it out. If not, keep it.
So we begin with $v_1$. Moving on to $v_2$, we see that $v_2$ is not in the span of $v_1$. So we keep it. Moving on to $v_3$, we see that $v_3$ is not in the span of $v_1$ and $v_2$, ie. $v_3 \notin \text{span}\left\{v_1, v_2 \right\}$. So keep this one too. And so on. Moving along in this fashion, you will at some point throw out vectors that are in the span of the ones previously considered, and by the end you will end up with five vectors that will include the three vectors in the set $S$ that you started with.
The details of this process are outlined in, for example, Axler's Linear Algebra Done Right, (section 2.B, pg. 39 in the 3rd edition).